The one-point connectification conjecture for locally connected $T_3$-spaces

Let $i=3$. A one-point connectification of a space $X$ is a connected space obtained by adjoining one point to $X$ as a subspace. A locally connected $T_i$-space has a $T_i$ one-point connectification if and only if it has no compact (or almost compact) component.

One-point connectification conjecture. Let $i=3$. A locally connected $T_i$-space has a $T_i$ one-point connectification if and only if it has no compact (or almost compact) component.

This conjecture extends the preceding characterization for $i=2$ to the excluded case $i=3$, corresponding to the stated duality between the earlier theorems. Its resolution is not given in the source.