Connected sum formula for ECH spectral invariants

Let $(Y_1,\lambda_1)$ and $(Y_2,\lambda_2)$ be closed connected contact three-manifolds with nonzero contact invariants, and let $k$ be a nonnegative integer. Write $(Y_1\#_{R}Y_2,\lambda_1\#_{R}\lambda_2)$ for their connected sum with parameter $R$, and let $c_k$ denote the $k$-th ECH spectral invariant. Connected sum conjecture.

$$\lim_{R\to 0}c_k((Y_1\#_{R}Y_2,\lambda_1\#_{R}\lambda_2))=\max\{c_i(Y_1,\lambda_1)+c_j(Y_2,\lambda_2)\mid i+j=k\}.$$

This conjecture predicts a max-convolution formula for ECH spectral invariants under connected sum, refining the connected sum computation of ECH at the module level. The statement is presented as a direction for future work, and its status is not resolved in the supplied text.