Is there a finite list of identites in the language of $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm})$ that generates all the others?

Question

Let $\Phi$ denote the set of all identities satisfied by $(\mathbb{N},0,1,+,\times,\mathrm{gcd},\mathrm{lcm}).$

Question. Is $\Phi$ finitely axiomatizable? If so, I'd like to see a list of identities.

Noteworthy elements of $\Phi$:

• $(\mathbb{N},0,1,+,\times)$ is a commutative semiring
• $(\mathbb{N},1,0,\mathrm{gcd},\mathrm{lcm})$ is a bounded distributive lattice (with bottom $1$ and top $0$)
• $\mathrm{gcd}(a,b+a) = \mathrm{gcd}(a,b)$
• $\gcd(a+b,\operatorname{lcm}(a,b))=\gcd(a,b)$
• $\mathrm{gcd}(a,b)\mathrm{lcm}(a,b) = ab$

The second last identity above actually follows from the preceeding one's; see Bill's answer here.