Maybe - there is preliminary evidence that supports it.
The Economist article is referring to this very recent working paper:
They found that:
when looking at enacted laws, the impact of mass shootings depends on the party in power. A mass shooting increases the number of enacted laws that loosen gun restrictions by 75% in states with Republican-controlled legislatures. We find no significant effect of mass shootings on laws enacted when there is a Democrat-controlled legislature.
They looked up to one year after the mass shooting for any effect on the legislation changes within the same state as the mass shooting. They compared it to to states in which no mass shootings had occurred (after controlling for some potential confounding factors).
By assuming that the occurrences are "salient and plausibly random occurrences" and by adding controls for the confounding factors that are time varying (unemployment, divorce rates, legislature size and restrictions, political parties, etc.) they claim that their study demonstrates causality. This is reasonable only if their assumption is correct, and they can successfully identify all of the important confounding factors.
The findings are discussed in a more readable version in a Harvard Business School web article. [Please be aware of the affiliations here: this is basically a press release version.]
This has not yet passed peer-review nor garnered post-publication review/criticism. This may give a skeptic some pause about whether to accept the claim yet. It is preliminary evidence to support the claim, but further peer-review to identify flaws and biases is yet to be completed.
Note: We've seen in previous questions that the definitions of mass shooting vary and affect the conclusions. The definition of "mass shooting" they use is:
as an incident in which four or more people, other than the shooter(s), are unlawfully fatally shot in a single incident not related to gangs, drugs, organized crime, or domestic disputes.