Positive-cone and optimality conjecture for the general first Physarum ansatz

Let $X(t)$ be the trajectory of the general first ansatz for the semidefinite program, and write $X(0)\succ F$ when $X(0)-F$ is positive definite. Let $F$ be a feasible solution of the SDP. General first-ansatz conjecture. If $X(0)\succ F$ for some feasible solution $F$, then the first ansatz stays in the positive definite cone and converges to the optimum of the SDP. The preceding theorem establishes positive-cone preservation under the additional assumption that $C^{-1}$ is linearly feasible; this conjecture asserts the stated behavior without that assumption.