A Bounded Convex Hilbert Domain with Vanishing Squeezing

We construct a bounded convex balanced domain in the complex Hilbert space C direct-sum ell^2 that is not holomorphic homogeneous regular. The domain is Omega = {(z,w): |z| + ||w||_infinity < 1, ||w||_2 < 1}. Its squeezing function at (t,0) is bounded above and below by positive constant multiples of 1-t as t increases to 1. The upper bound follows from explicit holomorphic retractions onto polydiscs of increasing dimension and an elementary Hilbert-space Schwarz argument. This gives a negative answer to Question 5 in Kim's 2018 Oberwolfach report. No boundary smoothness is claimed, and the separate Question 6 about squeezing constants of arbitrary bounded homogeneous domains is not addressed. This is an AI-assisted, self-audited, unrefereed preprint; no external review, formal verification or absolute priority is claimed.