Erlandsson–Parlier conjecture on self-intersection excess of systoles on compact hyperbolic surfaces

Let $\Sigma$ be a compact hyperbolic surface, and let $I_k(\Sigma)$ denote the maximal self-intersection number of a $k$-systole on $\Sigma$. Erlandsson–Parlier's conjecture. One has

$$\underset{k\rightarrow+\infty}{\limsup} \hspace{0.2cm}(I_k(\Sigma)-k)=+\infty.$$

For hyperbolic surfaces with at least one cusp, known results give $I_k(\Sigma)=k$ for all sufficiently large $k$, whereas this conjecture predicts unbounded excess self-intersection along a subsequence for compact surfaces.