GreenFunction for Helmholtz equation in arbitrary Rectangle region doesn't evaluate

Question

Bug persists through V13.0.0 or later

Here is a basic example found in the documentation of GreenFunction:

reg = Rectangle[{0, 0}, {1, 1}];
GreenFunction[
   {Laplacian[v[x, y], {x, y}] + v[x, y], 
    DirichletCondition[v[x, y] == 0, True]},
   v, Element[{x, y}, reg], {ξ, η}
]
(* Function[{x, y, ξ, η}, -4 Inactive[Sum][(Sin[π x K[1]] Sin[π ξ K[1]] Sin[π y K[2]] Sin[π η K[2]])/(-1 + π^2 K[1]^2 + π^2 K[2]^2), {K[1], 1, ∞}, {K[2], 1, ∞}]] *)

The above evaluates symbolically as expected.

But if I shift the region:

reg = Rectangle[{-1/2, -1/2}, {1/2, 1/2}]
GreenFunction[
   {Laplacian[v[x, y], {x, y}] + v[x, y], 
    DirichletCondition[v[x, y] == 0, True]},
   v, Element[{x, y}, reg], {ξ, η}
] 

then GreenFunction no longer evaluates and returns the input!

What could be the reason ? Is there a workaround?