Equivalence of Matrix Determinant Maximization Problems?

Question

I am interested in an optimization problem involving parameters $\boldsymbol{\mu}$ and $\boldsymbol{\pi}$, both strictly positive stochastic vectors, and variable $n \times n$ matrix $W$: \begin{equation} \max\{ |\det(W)|: \boldsymbol{\mu}^\top = \boldsymbol{\pi}^\top W, W \boldsymbol{1} = \boldsymbol{1}, W \in \mathbb{R}_+^{n \times n} \}. \tag{1} \end{equation} Here, $\mathbb{R}_+$ denotes the nonnegative reals. The constraints of this problem reflect that $W$ is right stochastic and that its transpose maps $\boldsymbol{\pi}$ to $\boldsymbol{\mu}$. The constraints are linear, but the "max absolute determinant" objective function is obviously nonconvex.

Given that $|\det(W)| = \big| \prod_{i=1}^n \lambda_i \big|$ and $\det(W W^\top) = \big( \prod_{i=1}^n \lambda_i \big)^2$, it seems to me that this problem would share its set of optimal solutions with \begin{equation} \max\{ \det(W W^\top): \boldsymbol{\mu}^\top = \boldsymbol{\pi}^\top W, W \boldsymbol{1} = \boldsymbol{1}, W \in \mathbb{R}_+^{n \times n} \}. \tag{2} \end{equation} Though the objective function in $(2)$ is still nonconvex, $W W^\top$ is at least positive semi-definite. So, owing to the monotonicity of $\log$, it seems to me that \begin{equation} \max\{ \log \det(W W^\top): \boldsymbol{\mu}^\top = \boldsymbol{\pi}^\top W, W \boldsymbol{1} = \boldsymbol{1}, W \in \mathbb{R}_+^{n \times n} \} \tag{3} \end{equation} would also have the same set of optimal solutions. This would be useful since $\log\det(\cdot)$ is concave over the set of positive definite matrices thus making $(3)$ a linearly constrained convex optimization problem.

Questions:

1. Is my reasoning correct? Basically, does $W^*$ optimal in $(3)$ imply $W^*$ optimal in $(1)$?
2. If not, would additionally imposing the condition $W W^\top \succ 0$ in each problem change the answer to my first question? I realize that if $W$ has one or more $\lambda_i = 0$, then the objective value of $(3)$ would be undefined (but the objective value of $(1)$ and $(2)$ would be zero). However, I have the analytical form of a feasible $\widehat{W}$ for which $\widehat{W} \widehat{W}^\top \succ 0$, so for this particular problem it seems explicitly imposing $W W^\top \succ 0$ does not affect the set of optimal solutions (though it would assuredly affect the set of feasible solutions).

Thanks in advance!

Edit: Since posting the question, I have realized that $\log\det(W W^\top)$ is concave in $W W^\top$. It is not concave in $W$. As such, whether or not $W^*$ optimal in $(3)$ implies $W^*$ optimal in $(1)$ is moot. That said, I am interested to know if there is a convex reformulation of $(1)$.