Maybe this should be one bug report and one feature request, but for now, I'll report them together. Consider the following program:

{-#LANGUAGE PartialTypeSignatures #-}
{-#LANGUAGE NamedWildCards #-}
{-#LANGUAGE NoMonomorphismRestriction #-}

foo :: _ => _outer
foo x = round $ (undefined::_inner) (1 + x)

This produces the following output in GHCi:

Foo.hs:5:8: Warning:
    Found hole ‘_’ with inferred constraints: (Integral b, Num a)
    In the type signature for ‘foo’: _ => _outer

Foo.hs:5:13: Warning:
    Found hole ‘_outer’ with type: a -> b
    Where: ‘b’ is a rigid type variable bound by
               the inferred type of foo :: (Integral b, Num a) => a -> b
               at Foo.hs:6:1
           ‘a’ is a rigid type variable bound by
               the inferred type of foo :: (Integral b, Num a) => a -> b
               at Foo.hs:6:1
    In the type signature for ‘foo’: _ => _outer

Foo.hs:6:29: Warning:
    Found hole ‘_inner’ with type: a -> Double
    Where: ‘a’ is a rigid type variable bound by
               the inferred type of foo :: (Integral b, Num a) => a -> b
               at Foo.hs:6:1
    Relevant bindings include
      x :: a (bound at Foo.hs:6:5)
      foo :: a -> b (bound at Foo.hs:6:1)
    In an expression type signature: _inner
    In the expression: undefined :: _inner
    In the second argument of ‘($)’, namely
      ‘(undefined :: _inner) (1 + x)’
Ok, modules loaded: Main.

The inferred constraints for _ (which I can't give a name, unfortunately) and the type reported in place of _outer are exactly what I expected. The type for _inner surprises me. Okay, the type is ambiguous, so for anything as general as undefined, arguably, it would need to default to something.

Let's use a typed hole instead of undefined:

foo :: _ => _outer
foo x = round $ _hole (1 + x)

gives

Foo.hs:6:17:
    Found hole ‘_hole’ with type: a -> Double
    Where: ‘a’ is a rigid type variable bound by
               the inferred type of foo :: a -> b at Foo.hs:6:1
    Relevant bindings include
      x :: a (bound at Foo.hs:6:5)
      foo :: a -> b (bound at Foo.hs:6:1)
    In the expression: _hole
    In the second argument of ‘($)’, namely ‘_hole (1 + x)’
    In the expression: round $ _hole (1 + x)
Failed, modules loaded: none.

Holy Guacamole, it still defaults to Double. I would consider this a bug, because GHC is telling me, that whatever I put there must result in a Double, which is too restrictive to be true. However, I seem to recall from the OutsideIn paper that there were some cases, even without GADTs, where principality was difficult. Was this one of them?

Moving on, I was actually trying to get all my holes typed for me in one compiler execution. GHC behaves according to spec; typed holes produce errors by default and when something breaks on an error, it doesn't produce the warnings for partial type signatures. Let's -fdefer-typed-holes and compile again:

Prelude> :set -fdefer-typed-holes
Prelude> :r
[1 of 1] Compiling Main             ( Foo.hs, interpreted )

Foo.hs:5:8: Warning:
    Found hole ‘_’ with inferred constraints: ()
    In the type signature for ‘foo’: _ => _outer

Foo.hs:5:13: Warning:
    Found hole ‘_outer’ with type: a -> b
    Where: ‘b’ is a rigid type variable bound by
               the inferred type of foo :: a -> b at Foo.hs:6:1
           ‘a’ is a rigid type variable bound by
               the inferred type of foo :: a -> b at Foo.hs:6:1
    In the type signature for ‘foo’: _ => _outer

Foo.hs:6:17: Warning:
    Found hole ‘_hole’ with type: a -> Double
    Where: ‘a’ is a rigid type variable bound by
               the inferred type of foo :: a -> b at Foo.hs:6:1
    Relevant bindings include
      x :: a (bound at Foo.hs:6:5)
      foo :: a -> b (bound at Foo.hs:6:1)
    In the expression: _hole
    In the second argument of ‘($)’, namely ‘_hole (1 + x)’
    In the expression: round $ _hole (1 + x)
Ok, modules loaded: Main.

Surely, this must be wrong. Suddenly () is the inferred set of constraints and an unconstrained a -> b will do for _outer. I would argue that the 1 + still demainds Num a and that round still requires an instance of Integral b, even if round's RealFrac constraint is satisfied by _hole producing a Double.

As said, maybe the erroneous types reported when -fdefer-typed-holes are a separate issue from the type of _hole not being the principal type, but I'm unsure, so I reported them together.