Worldsheet string theory dual to free $N=4$ super-Yang-Mills
In plain words
Gauge theory with many colors and no interaction at all should still be equivalent to a string theory in curved space with strings of zero tension. Finding that string theory explicitly would show from first principles how gauge fields turn into strings.
Precise statement
Planar free $N=4\ \operatorname{SU}(N)$ super-Yang-Mills ($g_{\mathrm{YM}} = 0$, $N \to \infty$) should be dual to tensionless type IIB strings on $\mathrm{AdS}_5 x S^5$. For $\mathrm{AdS}_3 x S^3 x T^4$ with one unit of NS-NS flux this was established with a worldsheet theory reproducing the symmetric-orbifold CFT (Eberhardt, Gaberdiel, Gopakumar 2018-2019). Construct the $\mathrm{AdS}_5$ worldsheet theory and show that its genus-zero correlators equal the planar free-field correlators of all single-trace operators.
What would settle it
A worldsheet theory whose genus-zero correlators are shown to equal planar free $N=4$ correlators for all single-trace operators, for example by localization of the worldsheet path integral onto free-field Wick contractions.
Status in the literature
Gaberdiel and Gopakumar (2021, arXiv:2104.08263) proposed a worldsheet description whose spectrum matches planar free $N=4$ super-Yang-Mills; correlators have not been derived from it.