Determine the optimal additive constant for the epsilon equals one-sixth upper bound

Determine whether the smallest possible integer constant A in the bound hdepth(S_n/I_n) ≤ ⌈n/2⌉ + ⌊n/6⌋ + A − 2 for all n ≥ 2 is A = 2, where S_n = K[x_1, …, x_n, y] and I_n = (x_1y, …, x_ny) is the edge ideal of the star graph with n rays.

Background

Theorem 2.6 establishes an upper bound of the form hdepth(S_n/I_n) ≤ ⌈n/2⌉ + ⌊εn⌋ + A(ε) − 2 for every ε with 0 < ε < 1/2, where the proof supplies an explicit, but potentially nonoptimal, value of A(ε). For ε = 1/6, the resulting value is A = 6, giving hdepth(S_n/I_n) ≤ ⌈n/2⌉ + ⌊n/6⌋ + 4.

The computer experiments reported in Example 3.1 indicate the stronger bound hdepth(S_n/I_n) ≤ ⌈n/2⌉ + ⌊n/6⌋ for all n ≥ 2. The authors state that the smallest possible value of the parameter A seems to be 2, leaving the optimal constant unresolved.