Let \(X\) be a hyperkähler variety of dimension four, namely a projective irreducible holomorphic symplectic fourfold. Let \(\iota\) be an anti-symplectic involution (i.e., an involution acting as \(-1\) on the symplectic form) of \(X\). In this situation, Bloch’s conjecture predicts that \(\iota^*=-\mathrm{ id}\) on \(A_{(2)}^i(X)\) for \(i=2,4\), and \(\iota^*=\mathrm{ id}\) on \(A_{(4)}^4(X)\). Here \(A_{(*)}^*(X)\) denotes the piece of bigrading, which is assumed to exist and is isomorphic to the graded piece \(\mathrm{ Gr}_{F}^jA^i(X)\) for the conjectural Bloch-Beilinson filtration. The main result of the present paper establishes a weak form of the conjecture for a 19-dimensional family of hyperkähler foufolds. Let \(X\) be the Hilbert scheme \(S^{[2]}\), where \(X\) is a very general K3 surface of degree \(d=10\). Let \(\iota\in\mathrm{ Bir}(X)\) be the anti-symplectic involution constructed by K. G. O’Grady [Geom. Funct. Anal. 15, No. 6, 1223–1274 (2005; Zbl 1093.53081)]. Then we have \(\iota^*=\mathrm{ id}\) on \(A_{(0)}^4(X)\), \(\iota^*=-\mathrm{ id}\) on \(A_{(2)}^4(X)\), \((\Pi_2^X)_*\iota^*=-\mathrm{ id}\) on \(A_{(2)}^2(X)\), and \((\Pi_4^X)_*\iota^*=\mathrm{ id}\) on \(A_{(4)}^2(X)\), where \(\{\Pi_j^X\}\) is a multiplicative Chow-Künneth decomposition, which is shown to exist in [M. Shen and C. Vial, The Fourier transform for certain hyperkähler fourfolds. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1386.14025)].