A recurrence-product conjecture for parametrized Euler sums

Define a sequence of polynomials $a_n=a_n(t)$ for positive integers $n$ by $a_1=a_2=t^3$ and

$$n(n+1)^2a_{n+2}=n(2n+1)a_{n+1}+(n^3+(-1)^{n+1}t^3)a_n,\qquad n\geq 1.$$

Recurrence-product conjecture. The sequence satisfies

$$\lim_{n\to\infty}a_n=t^3\prod_{n=1}^\infty\left(1+\frac{t^3}{8n^3}\right).$$

This is presented as an intriguing reformulation of the parametrized alternating Euler-sum identity discussed immediately beforehand. The supplied text gives numerical evidence for the underlying identity but does not state whether this recurrence formulation has been proved.