If the equation $x_i$-subproblem showed below is not strictly convex
$\arg \min_{x_i}=f_i(x_i)+\frac{\rho}{2}\|A_ix_i+\sum_{j\neq i}A_jx_j^k-c-\frac{\lambda^k}{\rho}\|_2^2$
Why adding the proximal term $\frac{1}{2}\|x_i-x_i^k\|^2_{P_i}$ showed below can make the subproblem strictly or strongly convex?
$arg \min_{x_i} f_i(x_i)+\frac{\rho}{2}\|A_ix_i+\sum_{j\neq i}A_jx_j^k-c-\frac{\lambda^k}{\rho}\|_2^2+\frac{1}{2}\|x_i-x_i^k\|^2_{P_i}$
The sum of a convex function and a strongly convex function is again strongly-convex. This is a direct application of the definition of strong-convexity.
On the other hand, it should be clear that $u \mapsto u^TP_iu$ is strongly convex if $P_i$ is positive definite.