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"Mandatory"? No, but when working with multiple BGP gateway AS routers, and IGP defaults, you got to be careful you don't create routing loops between the two protocols.
An example of such a problem situation, suppose your IGP default points to BR1, but once traffic gets there, BR1 wants to send the traffic out to BR2. If your internal topology is such it believes the best way from BR1 to BR2 is starting with switch 1 (assuming it's routing), then switch 1 will send the traffic back to BR1, and you have a routing loop.
If BR1 believes the best path to BR2 is via the direct link between them, as shown on your diagram, you're okay. But what if that link fails?
There are several methods to avoid the routing loop situation, one being having any transit routers, between BR 1 and 2 also running iBGP. Another would be to inject the BGP routes into the IGP (generally not done, although a possible solution when working with non-Internet). Another might be to build some kind of tunnel between BRs 1 and 2 (GRE, MPLS tunnel). Another, would be to use a VLAN to have the two BRs be on the same internal network.
