Consider the following linear system of equations $$ A^\top y=(A^\top A) x $$ with unknown $x$, where $$ A\equiv \left[\begin{array}{ccc|cccc|cccc} 1 & 0 & 0 & 0 & p_1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & p_2 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & p_3 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & p_4 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & 0 & p_5 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0 & p_6 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & p_7 & 0 & 0 & 0 & 1 & 0 & 0\\ \hline 0 & 0 & 1 & 0 & 0 & 0 & p_8 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & p_9 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & p_{10} & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0& p_{11} & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & p_{12} & 0 & 0 & 0 & 1 & 0 & 0\\ \end{array}\right] $$
The $p$'s in the central block are strictly positive, but potentially equal among each other.
Question: I would like to find necessary and sufficient conditions such that the system has a unique solution wrto $x$.
In what follows, for every $t=1,...,12$, I will refer to $t_1$ as the position of the first "1" in row $t$ and $t_2$ as the position of the "p" in row $t$.
I have found some sufficient conditions:
(1) Anyone element of $x$ is set equal to a known number by the analyst. For example, $x_1=0$.
(2) For every $j\in \{4,5,6,7\}$, there are at least two rows $t,\tau$ such that $t_1=\tau_1$, $t_2=\tau_2=j$, and $p_t\neq p_{\tau}$.
Note that (2) is not necessary. For example, delete the $4$-th row of $A$ and impose (1). Then, the system has a unique solution.
I have found some necessary conditions.
Necessary conditions are:
(1) Anyone element of $x$ is set equal to a known number by the analyst. For example, $x_1=0$.
(2) For $j=4,5,6,7$, column $j$ has at least two distinct strictly positive elements.