Are two terms where one is without a $\lambda\beta$ normal form unconvertible in $\lambda\beta$?

Question

I know that if you try and make the theory

$$\lambda\beta+\{s = t\ |\text{ s, t are terms without }\lambda\beta\text{ normal forms}\}$$

then that theory becomes inconsistent. Are two terms where one is without a $\lambda\beta$ normal form also unconvertible in $\lambda\beta$, ie can it ever be true that $\lambda\beta \vdash s=t$ if $s$ dosen't have a normal form?

Answer 1

It is trivially the case that two terms can be $\beta$-convertible even though they do not have a normal form.

Consider:

$$((\lambda x.x)\ (\lambda x.x\ x))\ (\lambda x.x\ x)$$

and

$$(\lambda x.x\ x)\ (\lambda x.x\ x).$$

In one step we have $$((\lambda x.x)\ (\lambda x.x\ x))\ (\lambda x.x\ x)\to (\lambda x.x\ x)\ (\lambda x.x\ x)$$