left-eigenvector of $\Phi_{\rm p}^{-1}$ with eigenvalue $1$, and so it is the first row of $P$ when the eigenvalues are in descending order. Left-eigenvectors of a matrix are right-eigenvectors of the transpose of the matrix, so we have $$\begin{split}
        (\Phi_{\rm p}^{-1})^T\ket{v}&=A^{-1}G\ket{v}=A^{-1}GA^{-1}A\ket{v}\\
        &=A^{-1}\Phi_{\rm p}^{-1}A\ket{v}=\lambda\ket{v}\;,\\
        \implies\Phi_{\rm p}^{-1}&A\ket{v}=\lambda A\ket{v}\;,
        \end{split}$$ where $\bra{v}$ is an arbitrary left-eigenvector of $\Phi_{\rm p}^{-1}$. Combined with our choice of scale for the first row of $P$, we conclude that the first column of $P^{-1}$ is $(h_1,h_2,\dots,h_{d^2})^T$.

Now suppose $\Phi_{\rm p}$ is such that $\lambda(\Phi_{\rm p})=\lambda(\Phi_{\rm SIC})$. Then $G=P^{-1}DPA$ where $[D]_{ij}=\frac{1}{d+1}(\delta_{ij}+d\delta_{i1}\delta_{j1})$, and $$\begin{aligned}
\label{gram}
                    [G]_{ij}&=&\sum_{klm}[P^{-1}]_{ik}[D]_{kl}[P]_{lm}[A]_{mj}\nonumber\\
            &=&\sum_{klm}[P^{-1}]_{ik}\left[\frac{1}{d+1}(\delta_{kl}+d\delta_{k1}\delta_{l1})\right][P]_{lm}\delta_{mj}h_m\nonumber\\
                        &=&\frac{1}{d+1}\sum_{kl}[P^{-1}]_{ik}(\delta_{kl}+d\delta_{k1}\delta_{l1})[P]_{lj}h_j\nonumber\\
            &=&\frac{1}{d+1}(h_j\delta_{ij}+dh_j[P^{-1}]_{i1}[P]_{1j})\nonumber\\
            &=&\frac{1}{d+1}(h_j\delta_{ij}+h_ih_j)\;.
        \end{aligned}$$ In the last step we used that $[P]_{1j}=1/d$ and $[P^{-1}]_{i1}=h_i$. If this Gram matrix comes from a MIC, one may use $$[G]_{ii}=h_i^2\tr\rho_i^2=\frac{1}{d+1}(h_i+h_i^2)\;,$$ and the fact that $\tr\rho_i\leq1$ to show that $h_i\geq 1/d$. As the average $h_i$ value must be $1/d$, this implies that $h_i=1/d$ for all $i$ and furthermore that each $\rho_i$ is rank-$1$. Substituting this into Eq.  gives $$[G]_{ij}=\frac{d\delta_{ij}+1}{d^2(d+1)}\;,$$ that is, the MIC is a SIC and $\Phi_{\rm p}=\Phi_\text{SIC}$.

Let $s(A)$ denote the vector of singular values of the matrix $A$ in nonincreasing order. The proof of the following lemma may be found in Appendix A.

\[maj\] Let 