Let me help you determine how you would answer your own question.
The way is to read the paper, and in particular, look at the theorems and the definitions of the notation and the assumptions. Don't just read the abstract. You have to read the whole paper.
For instance, you can look at Theorem 1.1, noticing the phrase "...for some $n$ and $t$..." -- to me, that suggests that $t$ is a constant in Theorem 1.
Or, you could look at Theorem 3.1, which explicitly mentions that $t$ is polynomial in $n$, where $n$ is the length of the SAT-instances, in the context of Theorem 3.1.
It's also often helpful to go back to the original paper that introduced the conjecture, and see how they formulated it. In this case, you can learn (from reading the introduction to the paper you link to) that Bodlaender  was the originator of the conjecture, and Fortnow and Santhanam  proved the conjecture under a certain assumption. The paper you link to (the Dell paper) might be assuming papers have read those prior work and are familiar with the technical details of the formulation of the conjecture. So, go read those papers, too.
Bottom line: you can't stop at reading just the abstract or just the first sentence of the introduction. You may have to dive into the details, and even into prior papers, to resolve questions like this. These papers are written for other researchers (who are comfortable doing all that).