The bypass-map compatibility conjecture for surgery cobordisms

Let $K\subset S^3$ be a knot, let $S^3_{-n}(K)$ be obtained by $(-n)$-surgery, and let $W$ be the natural cobordism from $S^3$ to $S^3_{-n}(K)$. Let $t_i:H_2(W_{-n})\to\mathbb{Z}$ be the basic-class label appearing in the decomposition of the cobordism map, and let $\mathcal{I}_+$ denote the relevant summand in the decomposition of framed instanton Floer homology. Bypass-map compatibility conjecture. There exists an integer $N$ such that, for every integer $i\in[0,n-1]$, under the identifications

$$I^{\sharp}(-S^3_{-n}(K))\cong\mathcal{I}_+(-S^3_{-n}(K),\widehat{K}),\qquad I^{\sharp}(-S^3)\cong\mathcal{I}_+(-S^3_{1/0}(K),K),$$

we have

$$I^{\sharp}(W_{-n},t_i)=I^{\sharp}(W_{-n})\big|_{\mathcal{I}_+(-S^3_{-n}(K),\widehat{K},N-i)},$$

and this cobordism map can be recovered by bypass maps. This conjecture concerns compatibility between the decomposition of the surgery cobordism map and the decomposition of $I^{\sharp}(-S^3_{-n}(K))$. Its resolution status is not specified in the supplied text.