Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$ 
I am interested in the following questions.

(1) Is the sentence $(\forall\bar x)(\exists\bar y)w=1$
true in every group if it is true in every finite group? 

(2) Is the sentence $(\exists\bar x)(\forall\bar y)w=1$  true in every group if it is true in every finite group? 

For $m=2$ and $n=4$ the answer to (1)
is negative, in general
 [T. Coulbois, A. Khelif,
Proc. AMS 127 (1999), No. 4, 963--965].
In  "On sentences true in all finite groups"
I asked the questions (1) and (2) for $m=n=1$.
Bjoern Kjos-Hanssen gave positive answers to these questions.
In fact, his answers can be generalized: 
an answer is positive for (1) if $m=1$, and for (2) if $n=1$.
The open questions I want to ask:

(a) Is  $(\forall xy)(\exists z)w(x,y,x)=1$
true in every group if it is true in every finite group? 

(b) Is  $(\exists x)(\forall yz)w(x,y,z)=1$  true in every group if it is true in every finite group?