No, it will not. In general, I do not think that you can dissolve HDPE in any "reasonable" solvent at room temperature. High temperatures usually have to be used (you can have a look at high temperature GPC of PE for inspiration).
it is not soluble since it is a low boiling temperature solvent, mainly the HDPE since it is highly crystalline polymer, high boiling aromatic solvent (naphta) solvents in conjonction with the high temperature may impart some plasticization (mainly at high concentration). Regards
While acetone will not dissolve HDPE, it can lead to material failure. If you leave acetone in a Nalgene bottle for long periods, it will eventually develop horizontal (i.e. circumferential) solvent-stress cracks at the level of the liquid-vapor interface, leading to failure when squeezed. (It is surprising how little vapor escapes through the nozzle at the end of the dispensing tube over the course of a year.) This frequently happens in our teaching laboratories if acetone-filled wash bottles are left partly filled from the end of one spring term to the beginning of another.
This problem does not occur when the wash bottles are in frequent use, since the liquid level constantly changes. It would not surprise me if larger institutions that offer organic chemistry every term do not notice this problem. Note that pure acetone is almost always sold only in glass or metal containers, most likely for this reason.
I do not know if other solvents (e.g. benzene, toluene) do the same thing, since we typically do not store them in Nalgene bottles for long periods.