OTS11 — Resume check - what state must be stored for each thread?

On page 10-40 of the OTS specification appears a description of the
checking to be performed on a Current::resume operation if checked
transactions are implemented.

Resume Check

Before allowing a client or object to associate a transaction context
with its thread of
control, a check is made to ensure that this transaction context was
previously
associated with the execution environment of the thread. This would be
true if the
thread either created the transaction or received it in a
transactional operation.

Does this means that an unrestricted amount of state must be retained
for the lifetime of each thread? That is, do all the transactions ever
created in a thread, and all the transactions ever received in a
transactional operation handled by that thread need to be stored so that
the Control argument to "resume" can be compared against each of them to
implement the check?

That can't be the case, but it that's how I'm interpreting it. What is
this really trying to say?