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Question 1: This is rather subjective depending what aspects of omni-channel are important to you. I am not sure if you have seen this Resource Center Article of ours discussing Salesforce Omni-Channel with Genesys Cloud: https://help.mypurecloud.com/articles/salesforce-omni-channel/
Question 2: Personally, and as an extension of the first question, the optimal solution for AXA would likely be using Genesys Cloud Omni-channel services within the Salesforce integration. Then your Agents need only work in the Salesforce environment with the Genesys Cloud embedded client. You can also configure status to sync so that your agents can receive interactions from both channels, if desired, without being over unitized. Genesys Cloud for Salesforce supports your inbound/outbound voice calls, callbacks, chat, emails, message interactions, and voicemails.
Question 3: Again the answer will be based off AXAs requirements, goals, and objectives. Genesys Cloud integrates well with Saleforce and you can use Data Actions to perform updates in both systems. So if you wanted the call routing metrics from Genesys Cloud but the contact cards to be updated in Salesforce, for example, then you could configure Genesys Cloud as your omni-channel provider and use scripts to load the contact cards in Salesforce for updating. There are many ways it can be configured. All metrics Genesys Cloud generates on interactions can be pulled via public API and used for reporting or dashboards within Salesforce, for example.
For the last three questions (4, 5 & 6) I can only really state the fairly obvious ones: The time and cost to first configure and then maintain these channels will be less if you do it all in one environment. Once in place the ongoing costs between would depend on your contract/usages costs with vendor. In regards to speed of connection and routing I'd imagine these are relative. Then depending on how you perform your reporting (both real-time and historic) if you use a third party reporting system based off API then there may take longer to build standard, comparable reports if you have mixed omni-channel, but if you are using standard reporting in either service then it would be simpler to configure, manage, and digest reporting if you used just the one service.
Genesys Cloud really specialises in Customer Experience and is constantly evolving to provide a more tailored experience for your Customers and staff too. This should be factored in to your decision making and, though less quantifiable than initial and ongoing costs, Customer experience and satisfaction will offset these costs in renewed business.
I hope some of this feedback has been helpful.
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