  Standard tableaux of skew shape are fundamental objects in enumerative and
algebraic combinatorics and no product formula for the number is known. In
2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of
products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a
$q$-analogue of this formula in terms of skew semistandard tableaux (SSYT).
They also showed, partly algebraically, that the Hillman--Grassl map,
restricted to skew semistandard tableaux, is behind their $q$-analogue. We
study the problem of circumventing the algebraic part and proving the bijection
completely combinatorially, which we do for border strips. For a skew shape, we
define minimal semistandard Young tableaux, that are in correspondence with
excited diagrams via a new description of the Hillman--Grassl bijection and
have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with
the terms of the Okounkov-Olshanski formula (OOF) for counting standard
tableaux of skew shape. Our construction immediately implies that the summands
in the NHLF are less than the summands in the OOF and we characterize the
shapes where both formulas have the same number of summands.
