Investigating a Minor Illusion uses an action, and the pursuers in a chase "are strongly motivated to use the Dash action every round." (DMG 252) Unless there's a compelling reason not to Dash, the pursuer shouldn't have a chance to do any investigating.
Whether there's a compelling reason is dependent on the situation and how the DM envisions the setting of the chase. If your pursuer is just 10 feet behind you and rounds a corner to find an alley with 30-foot walls that's completely empty except for one 5-foot crate, that pursuer is realistically motivated to take a look at that chest.
(But the DMG assumes that chases take place either in generic cities or in generic wilderness, and in cities where there's always a 5% chance you'll run into a beggar and a 5% chance you'll run into a pack of dogs, it can be assumed by the laws of cinema that whatever alley you turn down is full of crates, or barrels, or piles of junk that you can blend into.)
Assuming your hiding place makes sense, under "Ending a Chase" (DMG 253) it says a chase "ends when one side or the other stops, when the quarry escapes, or when the pursuers are close enough to their quarry to catch it." When you hide inside your illusory crate, you stop. So the chase ends.
However, the next sentence is weird: "If neither side gives up the chase, the quarry makes a Dexterity (Stealth) check at the end of each round [...] The result is compared to the passive Wisdom (Perception) scores of the users." The phrasing is awkward, but you haven't "given up the chase" because you haven't fallen down exhausted or turned around to surrender.
The DMG lists various reasons for you to have advantage or disadvantage on your stealth check, and I think we'd all agree that being completely obscured by an illusory crate qualifies. (Depending on the situation, there may be a strong motivation for the DM to say you succeed automatically.) With passive Perception scores being what they are, and advantage being what it is, it's very likely that you'll succeed on this check, in which case "that quarry [who is you] escapes." The pursuers have failed, and would only bother taking a look at any individual crates if the DM were feeling especially sadistic.
If you fail your Stealth check, "the chase continues for another round." You're still not moving, though, so the chase immediately ends again—That doesn't make any sense. This is where the chase rules break down for our purposes, because you have effectively withdrawn from the paradigm of a chase. Your failed Stealth check has betrayed your general location, and now the pursuers are just plain old NPCs. They will or won't investigate your illusion based on the normal rules of Investigation checks, because the chase is has ended.