Gali and van Rens: The Vanishing Procyclicality of Labor Productivity
A napkin theoretical exploration of the issue, to find possible directions to look for explanations, could go like this:  

According to the paper, the volatility of output and productivity has fallen. So for the correlation to have gone down, the covariance must have fell steeply.
Modeling output as

$$Y = AK^a(h\cdot L)^{1-a}$$

where $h$ is the "effort per unit" variable, we have 

$${\rm Cov}(Y, Y/L) = E(Y^2/L) - E(Y)E(Y/L)$$

Now

$$Y^2/L =   A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}$$

$$Y/L =   A(K/L)^{a}\cdot h^{1-a}$$

So

$${\rm Cov}(Y, Y/L) = E\left[A^2(K/L)^{2a}\cdot L \cdot h^{2(1-a)}\right] -E\left[A(K/L)^{a}\cdot L\cdot h^{1-a}\right]E\left[A(K/L)^{a}\cdot h^{1-a}\right]$$

We observe ${\rm Cov}(Y, Y/L) \rightarrow 0$. Now, I am in the mood of assuming that the capital-labor ratio is essentially a constant, as is $A$. More over, that the effort-per-unit variable is independent of the amount of labor hours. Under this scenario, we have

$${\rm Cov}(Y, Y/L) = A^2(K/L)^{2a} \cdot E(L)\cdot \big[ E\left[(h^{1-a})^2\right] -\left[E(h^{1-a})\right]^2\big]$$

$$\implies {\rm Cov}(Y, Y/L) \rightarrow 0 \implies {\rm Var}(h^{1-a}) \rightarrow 0$$

So here, the effort variable should be approximately a constant too.

--

In a more agnostic approach, using lower case letters to denote logarithms, we would have

$${\rm Cov}(y, y-\ell) = Var(y) - {\rm Cov}(y, \ell)$$

If what we observe is

$${\rm Cov}(y, y-\ell) \approx 0 \implies  Var(y) \approx {\rm Cov}(y, \ell)$$

But this means that in a least-squares regression of $\ell$ on a constant $\gamma_0$ and $y$
$$\ell = \gamma_0 + \gamma_1y + u$$
we would obtain $\gamma_1 =1$ and the relation

$$\hat \ell = \hat \gamma_0 + y \implies \approx Y = e^{-\gamma_0}L$$

(with $\gamma_0 &lt;0$ most probably). This is consistent with the previous approach since the constant $e^{-\gamma_0}$, may be seen as containing the $A(K/L)^ah^{1-a}$ terms, which were assumed or emerged as constants for the result of zero-covariance to occur. Here, there is more flexibility since we only require that their product is constant, not each one separately.