Conjecture on higher normalized eigenvalues of the projective plane

Let $\Lambda_k(\mathbb{RP}^2)$ denote the supremum of the normalized $k$-th Laplace eigenvalue over metrics on the real projective plane. A sequence of metrics is said to degenerate to a touching union when it converges to a configuration of components meeting at points. Projective-plane eigenvalue conjecture. For every $k\geqslant 1$,

$$\Lambda_k(\mathbb{RP}^2)=4\pi(2k+1).$$

For $k\geqslant 2$, the supremum cannot be attained by a smooth metric and is realized in the limit by a sequence degenerating to a union of $k-1$ identical round spheres and a standard projective plane touching one another, with the ratio of the area of the projective plane to the area of each sphere equal to $3:2$. This extends the known value $\Lambda_2(\mathbb{RP}^2)=20\pi$ and predicts the degeneration pattern for all higher indices.