Multiplicity bound from containment of powers

Suppose that $\mathfrak{a}$ and $J$ are $\mathfrak{m}$-primary ideals in a $d$-dimensional Noetherian local or Noetherian graded $k$-algebra $(R,\mathfrak{m})$, where $k$ is a field of arbitrary characteristic. Assume that $J$ is generated by a full system of parameters, homogeneous in the graded case. Multiplicity-containment conjecture. If $N\geq 0$ and

$$\mathfrak{a}^{N+1}\subseteq J,$$

then

$$e(\mathfrak{a})\geq\left(\frac{d}{d+N}\right)^d e(J).$$

This characteristic-independent problem was raised in HMTW and is stated in the paper as a conjecture; the paper notes that it would imply the preceding F-threshold multiplicity conjecture. No resolution is supplied.