The Naito--Sagaki conjecture asserts that the branching rule for the restriction of finite-dimensional, irreducible polynomial representations of $GL_{2n}(\mathbb{C})$ to $Sp_{2n}(\mathbb{C})$ amounts to the enumeration of certain ``rational paths'' satisfying specific conditions. This conjecture can be thought of as a non-Levi type analog of the Levi type branching rule, stated in terms of the path model due to Littelmann, and was proved combinatorially in 2018 by Schumann--Torres. In this paper, we give a new proof of the Naito--Sagaki conjecture independently of Schumann--Torres, using the branching rule based on the crystal basis theory for $\imath$quantum groups of type $\mathrm{AII}_{2n-1}$. Here, note that $\imath$quantum groups are certain coideal subalgebras of a quantized universal enveloping algebra obtained by $q$-deforming symmetric pairs, and also regarded as a generalization of quantized universal enveloping algebras; these were defined by Letzter in 1999, and since then their representation theory has become an active area of research. The main ingredients of our approach are certain combinatorial operations, such as promotion operators and Kashiwara operators, which are well-suited to the representation theory of complex semisimple Lie algebras.