Ehrhart formula conjecture for panhandle matroids

Let $a,b$ be positive integers. Let ${\texttt{Pan}}_{b+1,a+b,a+b+2}$ be the panhandle matroid, let $i(M,t)$ denote the Ehrhart polynomial of the matroid polytope of $M$, and let $U_{b,a+b+1}$ be the uniform matroid. Panhandle Ehrhart formula conjecture. One has

$$i({\texttt{Pan}}_{b+1,a+b,a+b+2},t)=i((1,1,a,b),t)=\left(\frac{a+1}{a+b+1}t+1\right)\cdot i(U_{b,a+b+1},t).$$

If true, this formula would manifestly show the Ehrhart positivity of the panhandle matroid ${\texttt{Pan}}_{b+1,a+b,a+b+2}$; the supplied text reports that its Ehrhart positivity was previously proved, but gives no resolution of the displayed formula.