Guys can Jenny's personal weakness Izzie be sorted by another investigator?

Although paragraph (1.2) on Triggered Abilities in FAQ implies it cannot be done, there is a weird wording in the Rules Reference, page 21 under "Weakness": "When an investigator draws a weakness with an encounter cardtype (for example, an enemy or a treachery weakness), resolve that card as if it were just drawn from the encounter deck."
That makes some people think that a Weakness card is an encounter card.

What is a weakness with an encounter cardtype? A cardtype on Izzy is Treachery for sure.

This makes me confused.

The short answer to the question is yes, Searching for Izzie can be dealt with by another investigator.

A weakness which is an asset (e.g. Daisy's Necronomicon), an event (e.g. Agnes's Dark Memory), or a skill (none at the moment) is a player card. Any weakness which is not a player card, which at the moment means it's either a treachery or an enemy, is an encounter card.

Since Searching for Izzie is an encounter card, any investigator who is at the same location is permitted to use its triggered ability as stated in section 1.2 of the FAQ.

For an in-depth discussion of the issue, which includes responses from the game designer, see this Board Game Geek thread: https://boardgamegeek.com/thread/1760182/abilities-other-players-weaknesses

If you go a bit further on page 21, you get this:

Quote

Weaknesses with an encounter cardtype are, like other encounter cards, not controlled by any player. Weaknesses with a player cardtype are controlled by their bearer.

This says those weaknesses are encounter cards. So they're definitely covered by the second bullet point in (1.2).

This got me thinking, if Jenny is eliminated, does Searching for Izzie remain in play and affect her if not resolved.

My reading says yes, but I wanted other takes.

Relevant RR 10:

The cards he or she controls in play and all of the cards in his or her out-of-play areas (such as hand, deck, discard pile) are removed from the game.

= Any card that player owns but does not control that is in play remains in play, but if that card leaves play it is removed from the game.

So my take is Skids weakness gets discarded, but since S4I is not under Jenny's control, it stays in.

2 minutes ago, Jobu said:

This got me thinking, if Jenny is eliminated, does Searching for Izzie remain in play and affect her if not resolved.

No, as soon as Jenny is eliminated the weakness triggers and is then removed from the game. This is step 0 of Elimination, as defined on page 2 of the FAQ (Rulebook Errata).

@Jobu Quote from the FAQ.

Quote

(v1.0) Rules Reference page 10, column 1, “Elimination”
Add the following, as step 0 (i.e. before step 1): “ 0. For the purpose of resolving weakness cards, the game has ended for the eliminated investigator. Trigger any “when the game ends” abilities on each weakness the eliminated investigator owns that is in play. Then, remove those weaknesses from the game. ”

Note that Skids doesn't control Hospital Debts either, because treacheries and enemies are always controlled by the scenario (the relevant quote is in my post just above yours). Before the FAQ, it would have been discarded in step 4 (and removed from the game as specified in step 1).