Logarithmic upper-bound conjecture for the gap in minimal isogeny degrees

Let $\delta(p)$ be the value associated with a prime $p$. The observed gap is

$$\left\lfloor\sqrt[3]{\frac{p}{2}}\right\rfloor-\delta(p).$$

Logarithmic-gap conjecture. One has

$$\limsup_{p\to\infty}\frac{\left\lfloor\sqrt[3]{\frac{p}{2}}\right\rfloor-\delta(p)}{\log p}<\infty.$$

This predicts that the gap grows at most on the order of $\log p$. The claim is motivated by computations for the ranges examined in the paper, but remains unproved.