tl;dr: Not so much as a conspiracy, but more of a line of ideological influence which can be traced back to Marx.
Yes and no. Marx's writings, or what I remember of them, don't necessarily focused in developing a cogent plan for cultural elements for the conditions of communism or even as a criticisms of the capitalism of his time. He really was more focused on the economic/philosophic/historic conditions of the matter, but I'll leave that there as it is out of the topic. What I can believe is that the writers that were inspired by Marx's ideas like those of the Frankfurt School, Gramsci, Foucault, Said, Derrida, etc, etc. did write bodies of work that told those interested that culture and the elements which compose it are is as much of a battleground as anything else: Language, religion, traditions, forms of communication, entertainment, education, sexuality, and about anything else really, is a potential tool that is used or to be for purposes of instruction about ideology.
If you like semantics I think that "cultural Marxism" is misnamed but it is an easy buzzword for people whose knowledge of communism is Marx and Stalin, so it makes at least some sense that it is what is called.
That being said, the phenomenon itself I think is a bit of mix bag. The natural counterpoint of capitalism is communism and those of us who live in a capitalist country often hear any criticism towards it as nothing more than commie cultural Marxist propaganda. The US in general has been afraid of anything related to communism since at least the 1920's as it very much would fucking wreck the country (as it did most that adopted it), and the people are very much trained to react to anything with ties to Marx's ideas as communism. There are segments of the population that have just replaced Communism/Communism for Cultural Marxism/Marxist as it sounds more intellectual, but is often used as meaninglessly as the term was used through the Cold War to attack anybody left of Reagan.
That being said, again, the current of though in much of the Western world has shifted left. Partly as a backlash against the states of the "Right" which were blamed for WW1 and WW2, and partly because of the rise of mass forms entertainment and communications which allowed certain artistic and scientific tendencies to be disseminated faster and wider than ever before. Those "somewhat" niche writers that inspired mostly academics and fellow writers have in these past decades found their ideas gestate into enough influence within the idea spreading circles that they have turned into policies or the unofficial pillars to build change around.
As for myself, capitalism isn't great but it still the best. Yet that doesn't mean it can't stand some criticism and the natural ideological source of criticism will be from those more ideologically aligned Marx. However, it is not ideal breaking down the pillars of society without replacing them with something just as strong. Maybe the foundation of man and woman, or the priority of protecting human life are outdated and gay, but what is replacing these ideas are nothing but ill pipe dreams are going to blown away either by their natural culminations, or their inability to exist outside of the sheltered conditions that Western capitalism provides to its population.