They do count the Olympic break. It is stated specifically that if a player got healthy from LTIR during a roster freeze, the team must immediately get under the cap anyway.

Syvret doesn’t fit that exception.

13.9 Bona Fide Long-Term Injury/Illness Exception Conditioning Loan. A Player
who is on the Bona Fide Long-Term Injury/Illness Exception as set forth in Article 50
may, with his consent, during the term of such Bona Fide Long-Term Injury/Illness
Exception (but in no event during the first fourteen (14) calendar days and six (6) NHL
Games), be Loaned on a Conditioning Loan (the “Bona Fide Long-Term Injury/Illness
Exception Conditioning Loan”) for a period not to exceed up to the longer of six (6) days
and three (3) games, solely for the purpose of determining whether the Player is fit to
play. If the Club determines that it needs more time to assess the Player’s fitness to play,
the Club may file a written request by facsimile with the Commissioner’s Office, with a
copy to the NHLPA, in accordance with Exhibit 3 hereof, to extend the Loan for an
additional two (2) games. The Commissioner, upon good cause, may approve the onetime
extension. The Commissioner’s approval shall not be unreasonably withheld. A
Player on a Bona Fide Long-Term Injury/Illness Exception Conditioning Loan will
continue to be listed on Injured Reserve and will not count against the Club’s 23-man
roster limit. The Club’s Bona Fide Long-Term Injury/Illness Exception will continue
until the Conditioning Loan ends, and his Paragraph 1 NHL Salary and Bonuses will
continue to count against the Club’s Upper Limit and the Players’ Share during such time.
The Commissioner may take whatever steps he deems necessary to investigate the
circumstances under which a Player is placed on a Bona Fide Long-Term Injury/Illness
Conditioning Loan. If he has reason to believe or determines that the Club has used the
Bona Fide Long-Term Injury/Illness Conditioning Loan to evade Re-Entry Waivers or
otherwise to Circumvent any provision of this Agreement, he may take other disciplinary
action against the Club as he deems appropriate. A Bona Fide Long-Term Injury/Illness
Conditioning Loan may be extended on one occasion. This procedure can only be used
once during each period of time that the Player is on a Bona Fide Long-Term Injury
Exception.

It would only get Syvret off the roster for six days, but that might be long enough to do the necessary deals, and it would give the team a chance to see if he can be effective at the AHL level.