Prove the conjectural connection-coefficient formula

Prove that the connection coefficient b_nu for the stationary harmonic Robin problem in an infinite wedge satisfies b_nu = 1/Gamma(1+nu) for every nu > 0, equivalently for arbitrary wedge opening angles alpha = pi/nu.

Background

For the special angles alpha = pi/n, the paper proves exactly that the connection coefficient is b_n = 1/n!. This result motivates the conjecture b_nu = 1/Gamma(1+nu) for arbitrary positive nu.

Finite-element computations for several noninteger values of nu strongly support the conjectured formula, but the paper does not provide a proof. Establishing the formula would rigorously validate the proposed apex prefactor for arbitrary wedge angles.