Product bounds conjecture for Hilbert-depth estimates

Let $s$ be a positive integer and let $k$ satisfy the lower bounds specified below. The products are defined by the displayed finite products over the indicated integer ranges.

Product bounds conjecture. The following inequalities hold:

(a)

$$\prod\limits_{j=1}^{2s} \left( 1 + \frac{2s+1}{k-j} \right) \leq 2,$$

for $k\geq \frac{(2s+0.5)^2}{\ln(2)}$.

(b)

$$\prod\limits_{j=1}^{2s+1} \left( 1 + \frac{2s+2}{k-j} \right) \leq 2,$$

for $k \geq \frac{(2s+1.5)^2}{\ln(2)}$.

(c)

$$\prod\limits_{j=0}^{2s-1} \left( 1 + \frac{2s}{k-j} \right) \leq 2,$$

for $k \geq \frac{4s^2}{\ln(2)}$.

(d)

$$\prod\limits_{j=2}^{2s+1} \left( 1 + \frac{2s+2}{k-j} \right) \leq 2,$$

for $k \geq \frac{(2s+1)^2}{\ln(2)}$.

These experimentally motivated inequalities are intended to provide the estimates needed for the paper's sharper bound concerning the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph. Their resolution would clarify the relevant asymptotic and finite-parameter bounds.