Strengthen Theorem \ref{thm-Domar-s^-b} to eliminate the multiplicative constant

Determine whether, under the hypotheses of Theorem \ref{thm-Domar-s^-b} (bounded open set Ω ⊂ R^k, 0 < p_* < k, closed B ⊂ R^k, compact p_*-admissible A ⊂ Ω with m(A)=0, and g(1/t) belonging to a Hardy field containing all power functions with g(t) ≤ C_1 t^{-β} on (0,1]), every subharmonic u on Ω \ B satisfying u(x) ≤ g(dist(x, A ∪ B)) for x ∈ Ω(A ∪ B) obeys the sharper bound u(x) ≤ g(v dist(x, B)) for some constant v ∈ (0,1) and all x ∈ Ω \ B.

Background

Theorem \ref{thm-Domar-s^-b} shows that, when A is p_*‑admissible and g has at most power-type singularity at 0^+, one can bound a subharmonic function u on Ω \ B as u(x) ≤ a g(v dist(x, B)) for some a > 1 and v ∈ (0,1). In special geometric settings (e.g., A a Lipschitz curve and g with weaker-than-any-power singularity), Theorem \ref{thm:mainP1} yields the stronger form u(x) ≤ g(C dist(x, B)).

The authors ask whether the multiplicative factor a > 1 in Theorem \ref{thm-Domar-s^-b} can be removed in the general p_*‑admissible setting, yielding a bound directly of the form g(v dist(x, B)).