The Eisenstein-family overconvergence-rate conjecture

Let $p$ be a prime. Let $b_{i,j}\in\mathbb Z_p[[q]]$ be the modular forms associated with the Katz expansion of $E^\ast_{\kappa}/V(E^\ast_{\kappa})$, and let $\nu_p$ denote the $p$-adic valuation of their coefficients. Define

$$d_p=\frac{p-1}{p(p+1)}.$$

Overconvergence-rate conjecture. For all $i,j\geq 0$,

$$\nu_p(b_{i,j})\geq d_pi-j.$$

Equivalently, with

$$\delta_p=\inf\left\{\frac{\nu_p(b_{i,j})+j}{i}\mathrel{\middle|}i\in\mathbb Z_{>0},\ j\in\mathbb Z_{\geq 0}\right\},$$

this predicts $\delta_p\geq\frac{p-1}{p(p+1)}$. The conjecture does not assert equality; computations for low primes find values of $i$ and $j$ attaining $d_p$, in which case it would imply $\delta_p=d_p$.