As addition to what I’ve argued in previous post (also see the connected thread on Abstraction and Given), here are quotes by but Frege and Husserl arguing that objectivity doesn’t entail existence/actuality.
From The Frege Reader, Michael Beaney (ed.), p96 (about The Foundations of Arithmetic)
Frege distinguishes what is objective (objectiv) from what is actual (wirklich), the later being the handleable (handgreflich), or spatial (räumlich), such that what is actual is only part of what is objective. Both the axis of the Earth and the centre of mass of the solar system are objective, but they are not actual like the Earth itself.
And Husserl explains that just because we think of something, it doesn’t mean that we should think that what we think of, has some intra-mental existence.
To assume that the intentional object is act-immanent, that is actually contained in the intention and therefore in possession of the same mode of being as the experience itself, leads to a rejection of the categorical distinction between act and object. That such a distinction does exist is easy to illustrate. (Hua 19/385)… (From Husserl’s Phenomenology, Dan Zahavi, p.15)
For what he means by “easy to illustrate”, see this post.
In this way Husserl shows that also we shouldn’t use the word “existence” for the objects of intention in some
different mode, e.g. “exist as thought”. That is… The thought about something exists, but the object of intention is not part of the thought.