Polynomial relationship conjecture for persistence entropy and shape factor

Let $M$ be a triply-periodic minimal surface. Write $\operatorname{PE}_1(M)$ for its 1-persistence entropy and $d(M)$ for its shape factor.

Polynomial relationship conjecture. For any triply-periodic minimal surface $M$, there exists a polynomial $P \in \mathbb{R}[t]$ such that

$$\operatorname{PE}_1(M) = P(d(M)).$$

The conjecture is motivated by experiments on the Gyroid and Schwarz surfaces, which provide evidence for polynomial relationships between the shape factor and 1-persistence entropy; the proposed exponential model was discarded because it offered little improvement over the polynomial model. Its general validity for all triply-periodic minimal surfaces remains unestablished.