About the Lindelöf degree of $(R) ^I$

Question

We already know that if $(R) ^ I$ is a Lindelöf space, then $I$ is countable. Is there a generalization of this proposition? If the Lindelöf degree of $(R) ^ I$ is a cardinal $\lambda$, it can be concluded that the cardinal of $I$ does not exceed $\lambda$? My first attempt was a generalization of the proof of the numerable case, but the classical argument rests in the fact that a regular Lindelöf space is automatically normal, and $(R) ^ I$ is not normal if $I$ is not countable, so a generalization in this way is not evident. Thank you for the further answers.

Answer 1

If $l(\mathbb{R}^I) = \kappa$, then I think $|I| \ge \kappa$ is pretty clear: if we'd have $|I| = \lambda < \kappa$, then $l(\mathbb{R}^I) \le w(\mathbb{R}^I) = |I|\cdot \aleph_0 = \lambda< \kappa$, which is a contradiction.

It wouldn't surprise me if $l(\mathbb{R}^I) = |I|$, for infinite $I$.