The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.

Suppose we pick $n$ values $x_1,\ldots,x_n$, independently sampled from $N(0,1)$ (mean 0, unit variance gaussian). Then, we form the (rank 3 at best) positive semidefinite matrix: $$A = \alpha ee^T + [\cos(x_i-x_j)],$$ where $e$ denotes the vector of all ones, and $\alpha > 0$ is a fixed scalar.

For $n \ge 3$, simple experiments lead one to conjecture that: $$e^TA^\dagger e = \alpha^{-1},$$ where $A^\dagger$ is the Moore-Penrose pseudoinverse of $A$ (obtained in Matlab using the 'pinv' function).

This should be fairly easy to prove with the right tools, such as a Matrix inversion lemma that allows rank deficient matrices or pseudoinverses. So my question is:

How to prove the above conjecture (without too much labor, if possible)?