  Fra\"iss\'e's conjecture (proved by Laver) is implied by the
$\Pi^1_1$-comprehension axiom of reverse mathematics, as shown by Montalb\'an.
The implication must be strict for reasons of quantifier complexity, but it
seems that no better bound has been known. We locate such a bound in a
hierarchy of Suzuki and Yokoyama, which extends Towsner's framework of partial
impredicativity. Specifically, we show that Fra\"iss\'e's conjecture is implied
by a principle of pseudo $\Pi^1_1$-comprehension. As part of the proof, we
introduce a cofinite version of the $\Delta^0_2$-Ramsey theorem, which may be
of independent interest. We also relate pseudo $\Pi^1_1$-comprehension to
principles of pseudo $\beta$-model reflection (due to Suzuki and Yokoyama) and
reflection for $\omega$-models of transfinite induction (studied by Rathjen and
Valencia-Vizca\'ino). In a forthcoming companion paper, we characterize pseudo
$\Pi^1_1$-comprehension by a well-ordering principle, to get a transparent
combinatorial bound for the strength of Fra\"iss\'e's conjecture.
