Initialization including equality constraints / zero cones

[mipals]

I have a question regarding the initialization step with regards to the equality constraints / zero cone. In the preprint (in the case of $P\neq 0$, but a similar thing can be said in the case of $P=0$) it is stated that the following linear system is solved

$$ \begin{bmatrix}P & A^\top \\ A & -I\end{bmatrix} \begin{bmatrix}x \\ z\end{bmatrix} = \begin{bmatrix}-q \\ b\end{bmatrix} $$

This would mean that the equality constraints are not necessarily satisfied (due to the presence of $I$). However, looking at the code the initialization seems to utilize the standard KKT-solver and even have a comment referring to the CVXOPT initialization

Clarabel.jl/src/solver.jl

Line 365 in dd709aa

# If there are only symmetric cones, use CVXOPT style initilization

Both cases point to towards that the equality constraints are actually satisfied. So I guess in practice you are solving something akin to

$$ \begin{bmatrix}P & A^\top \\ A & -I_E\end{bmatrix} \begin{bmatrix}x \\ z\end{bmatrix} = \begin{bmatrix}-q \\ b\end{bmatrix}, $$

where $I_E = I - E$ with $E$ being a matrix with ones on the diagonal entries corresponding to equality constraints?