( T1T_1First Countablekk-Menger ) ⇒ Hemicompact

See Proposition 5 of DOI 10.1016/j.topol.2005.07.015. In that proof, one can take On(K)O_n(K) to be X{xn(K)}X \setminus \{ x_n(K) \}, which is open by $T_1$.
The converse ( Hemicompact ⇒ ( T1T_1First Countablekk-Menger )) cannot be proven or disproven from other known theorems