Recurrence conjecture for the first two diagonal entries of the volume triangle

For $n\ge 3$, let $a_n$ be the first entry in the diagonal of the volume triangle, and for $n\ge 4$ let $b_n$ be its second entry. Let $V_n$ be the relative volume of $P_n$.

Recurrence conjecture for $a_n$ and $b_n$. The first diagonal entry satisfies

$$a_n=\frac{3V_n}{\binom{n}{2}}.$$

For $n\ge 4$, the second entries satisfy

$$(n-1)\left(\frac{b_{n+1}}{a_{n+1}}-\frac{b_n}{a_n}\right)=(n+2)\left(\frac{b_{n+2}}{a_{n+2}}-\frac{b_{n+1}}{a_{n+1}}\right).$$

The conjecture is motivated by numerical diagonal data for $n=3,\ldots,10$ and proposes formulas relating the diagonal entries to the relative volume and to one another. No proof or resolution is supplied in the paper.