Eisenbud–Goto conjecture for homogeneous prime ideals

Let $K$ be an algebraically closed field, let $R=K[x_1,\dots,x_N]$, and let $P\subset (x_1,\ldots,x_N)^2$ be a homogeneous prime ideal. Write $\operatorname{reg}(P)$ for its Castelnuovo–Mumford regularity, $e(R/P)$ for the multiplicity, and $\operatorname{ht}(P)$ for the height. Eisenbud–Goto conjecture. One has

$$\operatorname{reg}(P) \leq e(R/P)-\operatorname{ht}(P)+1.$$

The conjecture is refuted by counterexamples of McCullough and Peeva, who also showed that the regularity of $R/P$ cannot be bounded by any polynomial in terms of $e(R/P)$.