Do positive integers $a,b,c,d$ exist with the given properties?

Question

Inspired by this question : Amicable pairs of numbers and their product

I ask whether positive integers $a,b,c,d$ exist with the following properties :

$(1)\ \ \ \ \ 0<a<b<c<d\ \ \ $

$(2)\ \ \ \ \ ad=bc$

$(3)\ \ \ \ \ \sigma(a)=\sigma(d)$

$(4)\ \ \ \ \ \sigma(b)=\sigma(c)$

where $\sigma(n)$ denotes the divisor-sum-function. Upto $d=200$, there is no solution, so I conjecture that not all the conditions can be satisfied.

This would answer the question , whether distinct pairs of amicable numbers can have the same product , in a negative way.

Answer 1

$a=210$, $b=310$, $c=357$, $d=527$.