Simplify Gamma function given integer variables

I want to simplify the following

Gamma[1 + d - k]

given the constraint that 1 + d - k is a positive integer. I have tried using

FullSimplify[Gamma[1 + d - k], {1 + d - k ∈ Integers, 1 + d - k > 0}]

However, this simply returns Gamma[1 + d - k]. I am expecting (d - k)! to be returned. Is there another way to simplify this expression with Mathematica?

$Version
(* 12.1.0 for Mac OS X x86 (64-bit) (April 13, 2020) *)

FullSimplify[Gamma[1+ d - k],Element[d,Integers] && Element[k,Integers] && 1+d-k > 0]
(* (d-k)! *)