Let X be a simple random walk on Zdn with d≥3 and let tcov be the expected cover time. We consider the set of points Uα of Zdn that have not been visited by the walk by time αtcov for α∈(0,1). It was shown in [MS17] that there exists α1(d)∈(0,1) such that for all α>α1(d) the total variation distance between the law of the set Uα and an i.i.d. sequence of Bernoulli random variables indexed by Zdn with success probability n−αd tends to 0 as n→∞. In [MS17] the constant α1(d) converges to 1 as d→∞. In this short note using the Chen--Stein method and a concentration result for Markov chains of Lezaud we greatly simplify the proof of [MS17] and find a constant α1(d) which converges to 3/4 as d→∞.