  A recent result of Downarowicz and Serafin (DS) shows that there exist
positive entropy subshifts satisfying the assertion of Sarnak's conjecture.
More precisely, it is proved that if $y=(y_n)_{n\ge 1}$ is a bounded sequence
with zero average along every infinite arithmetic progression (the M\"obius
function is an example of such a \sq\ $y$) then for every $N\ge 2$ there exists
a subshift $\Sigma$ over $N$ symbols, with entropy arbitrarily close to $\log
N$, uncorrelated to $y$.
  In the present note, we improve the result of (DS). First of all, we observe
that the uncorrelation obtained in (DS) is \emph{uniform}, i.e., for any
continuous function $f:\Sigma\to {\mathbb R}$ and every $\epsilon>0$ there
exists $n_0$ such that for any $n\ge n_0$ and any $x\in\Sigma$ we have $$
\left|\frac1n\sum_{i=1}^{n}f(T^ix)\,y_i\right|<\epsilon. $$ More importantly,
by a fine-tuned modification of the construction from (DS) we create a
\emph{strictly ergodic} subshift, with all the desired properties of the
example in (DS) (uniformly uncorrelated to $y$ and with entropy arbitrarily
close to $\log N$).
  The question about these two additional properties (uniformity of
uncorrelation and strict ergodicity) has been posed by Mariusz Lemanczyk in the
context of the so-called strong MOMO (M\"obius Orthogonality on Moving Orbits)
property. Our result shows, among other things, that strong MOMO is essentially
stronger than uniform uncorrelation, even for strictly ergodic systems.
