GHC fails to infer types of certain valid mutually recursive bindings

Summary

User TophatEndermite on Reddit posted the following program and asked why GHC can't infer its type:

f = g 5 + g "hi"
g x = f * 0

This produces the following error:

T.hs:3:7: error:
    • No instance for (Num String) arising from the literal ‘5’
    • In the first argument of ‘g’, namely ‘5’
      In the first argument of ‘(+)’, namely ‘g 5’
      In the expression: g 5 + g "hi"
  |
3 | f = g 5 + g "hi"
  |       ^

But it should be quite easy to see that these bindings can be given sensible types:

f :: Integer
g :: a -> Integer

I thought the monomorphism restriction could be causing the inference failure, but turning it off doesn't help.

I also thought the Num constraint could be causing the failure, but even this program without constraints fails to be inferred by GHC:

plusInt :: Int -> Int -> Int
plusInt = (+)

timesInt :: Int -> Int -> Int
timesInt = (*)

zero :: Int
zero = 0

five :: Int
five = 5

f = plusInt (g five) (g "hi")
g x = timesInt f zero

Why can't GHC infer the types of f and g?

Environment

• GHC version used: 9.2.4