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In earlier work with C.~Monical, we introduced the notion of a K-crystal,with applications to K-theoretic Schubert calculus and the study of Lascouxpolynomials. We conjectured that such a K-crystal structure existed on the setof semistandard set-valued tableaux of any fixed rectangular shape. Here, weestablish this conjecture by explicitly constructing the K-crystal operators.As a consequence, we establish the first combinatorial formula for Lascouxpolynomials $L_{w\lambda}$ when $\lambda$ is a multiple of a fundamental weightas the sum over flagged set-valued tableaux. Using this result, we then provecorresponding cases of conjectures of Ross--Yong (2015) and Monical (2016) byconstructing bijections with the respective combinatorial objects.

K-theoretic crystals for set-valued tableaux of rectangular shapes