Polynomial transformations on block encodings

Let $P(x)=\sum_{j=0}^{L} c_j x^j$ be a degree-$L$ real polynomial with parity $L \bmod 2$ such that $|P(x)|\leq 1$ for all $x\in[-1,1]$. Suppose $U$ is a unitary and $\Pi,\widetilde{\Pi}$ are orthogonal projectors, and let $A=\widetilde{\Pi}U\Pi$. There exists a quantum circuit implementing a unitary $V$ such that $$P^{(\mathrm{SV})}(A)=(\bra{+}\otimes\Pi_L)V(\ket{+}\otimes\Pi),$$ where $\Pi_L=\widetilde{\Pi}$ when $L$ is odd and $\Pi_L=\Pi$ when $L$ is even. The circuit uses one ancilla qubit, $L$ total queries to $U$ or $U^\dagger$, $L$ queries to $\mathtt{C}_{\Pi}\mathtt{NOT}$, $L$ queries to $\mathtt{C}_{\widetilde{\Pi}}\mathtt{NOT}$, and $L$ controlled phase gates.