at this time vs. at that time

Try a search at
http://www.google.com
with:
site:bbc.co.uk "at this time"
and separately with:
site:bbc.co.uk "at that time"
(quotation marks are important)
which will perform a search on that bastion of good English, the BBC

You will find that "at that time" is related mostly with past tense, while "at this time" is mostly related to present tense (you're free to perform some stats and confront me on that, but ...). Thus your boss is, in general, correct.

However, you will find examples where the time is "localized" to the general time of the sentence, and "at this time" will be used for describing events which took place a long time ago:
--------
BBC - North East Wales Denbighshire history - Memories of ...

In 1954, as a 16 year old, I took up motorcycling and often visited Sun Garage motorcycle showroom and workshop and petrol outlet located on Regent Street opposite the junction with Church Street. These were exciting times for it was an opportunity to view the new machines in the showroom and chance to talk to Alan, Les, and Horace who I believe were partners in the business. It is amazing now to think that at this time petrol was dispensed from attended pumps located at the busy roadside. On one occasion sitting astride my BSA bantam motorcycle, Horace, a heavy smoker, served me with fuel. I recall I was horrified and too frightened to move or say anything, for Horace dispensed the fuel with a lighted cigarette dangling from his lips. I never ever went back there for fuel again.

http://www.bbc.co.uk/wales/northeast/sites/denbighshire/pages/int-eist1.shtml
--------

Thus time is relative, and an author can focus on the local time of the sentence and prefer/pretend to say "at this time," but see the preferences in the above.

This being a business letter, be conventional, as you want to minimize confusion.

N.B. Should you prefer AmE to BrE, make searches at the New York Times, with:

site:nytimes.com "at this time"

where the first hit gives something similar to your preference:
---------
But beginning in 2002, the United States branded North Korea as part of an axis of evil, threatened military action, ended the shipments of fuel oil and the construction of nuclear power plants and refused to consider further bilateral talks. In their discussions with me at this time, North Korean spokesmen seemed convinced that the American positions posed a serious danger to their country and to its political regime.

Jimmy Carter,  in the New York Times
http://tinyurl.com/gv2kq
----------

Could we widen the scope of possibilities to include at the same time and at the time?

I mention this because both of the original options seem to allow a alternative reading which implies that there were others moments in time when the same action was not performed, even though it could/should have been done.

Hi guys,

I'd like to offer an additional approach for you to consider in this excellent discussion.

In spatial terms, 'this' and 'that' relate to the concept of 'near' and 'far', eg this book vs. that book.

I have some of the same feeling about at this time and at that time.

At this time suggests to me some personal involvement, 'some closeness',  by the speaker. eg 'At this time, he confessed to me that he robbed the bank'.

At that time does not so strongly suggest to me personal involvement  by the speaker. eg 'At that time, he confessed to the police officer that he robbed the bank'.

It's just a subtle difference, not a clear and absolute rule.

Best wishes, Clive