Conjectured volume formula for parabolic-taxicab balls

Let $m$ and $r\ge 0$ be integers. In the parabolic-taxicab geometry, let $\mathscr{B}_{\mathrm{pc}}((m,m),r)$ denote the closed ball of center $(m,m)$ and radius $r$, and let $\mu$ denote its measure.

Parabolic-taxicab ball volume conjecture. The measure of this ball is

$$\mu\Big(\mathscr{B}_{\mathrm{pc}}\big((m,m),r\big)\Big) = \begin{cases} \frac{1}{12}\big(10 r^3+9r^2+26r+12\big), & \text{if $r$ is even;} \\[4mm] \frac{1}{12}\big(10 r^3+9r^2+26r+15\big), & \text{if $r$ is odd.} \end{cases}$$

The formula predicts the exact size of the metric balls arising from motion along the parabolic ladders together with unit horizontal and vertical jumps; the source presents it as a likely formula, and no resolution is supplied.