Projectile speed increase mastery not working with toxic rain.

The issue relates to toxic rain's dps being affected (or not?) by the mastery on the large bow wheel in the ranger's area, namely:

Increases and reductions to projectile speed also apply to damage with bows.

Discussed among the other's here:

https://www.pathofexile.com/forum/view-thread/3229132

The wording would suggest that the increases to proj. speed affects the damage with bows.
However, while damage with bows nodes affect TR's tooltip dps (it is visible as calculation increase), while this node taken in game does not (not visible as calculation increase).
People say, that this increase is visible in POB's calculation, however I cannot rely on this, as I cannot setup POB to import my char's, using it merely as a character's planner, besides I wouldn't count on sth that 3rd party program says, while the game's calculation contradicts it.

The wording suggests that this mastery should work if one actually has sources of proj. speed increases available, and I do have several.

IT DOES NOT.

So to reiterate:

The node's wording is clear.
The node is taken.
The generic damage with bows nodes increase the calculation in game.
The node despite it should because it equals the above, does not.

The support team refused to answer this question, giving the same canned response 3 times.

I equipped a character with Quill Rain and level 1 Toxic Rain, and had 240 tooltip DPS and 10.7 chaos damage per second. On allocating this mastery, the tooltip DPS went up to 317, while the chaos damage per second stayed at 10.7.

The mastery appears to work with Toxic Rain, scaling the damage of the skill's hits.

The mastery does not appear to apply to the chaos damage over time of Toxic Rain, which is expected because damage over time effects generally aren't scaled by modifiers that specificly apply to attacks, like "Damage with bows".

This interaction is further displayed on the passive tree by the fact that all passive nodes with "increased Damage with Bows" have a matching "increased Damage over Time with Bow Skills" line.

Checked this as well, it boosts damage of hits from Toxic Rain as expected.

"

So to reiterate:

The node's wording is clear.

So apparently not, when it goes like "something damage with weapon_name", it only implies hits with this weapon.