Yes, call the Citi Prestige concierge to make the booking. Although I haven't booked this specific rate yet through Citi, I'm assuming it's similar to what is available through Amex FHR or Virtuoso, and the rate you'll be offered will be the same as the best available flexible rate booked directly with the hotel.

The details from the T&C don't specifically mention RC or FS, but they are as follows:

SPECIAL BENEFITS AT LUXURY HOTELS PROVIDED BY TEN LIFESTYLE LTD. — TERMS AND CONDITIONS
To take advantage of the Special Benefits at Luxury Hotels, card members must (1) book through Ten Lifestyle Management Ltd., and (2) must reserve and pay in full for reservations, including deposit and final payment, with a valid CITI PRESTIGE credit card.

All travel must be booked and commenced before a benefit expires. To check a benefit's expiration, contact your Citi Prestige Concierge by calling the number on the back of your card.
All benefits and offers are subject to availability and are subject to change or cancellation without notice. Benefits may generally be redeemed multiple times.
Such benefits may not be combinable with a travel service provider's other offers.
Room upgrades and program amenities are based upon availability at time of arrival unless otherwise noted.
Room upgrades are offered only for designated hotels and room categories and individual hotel participation may be withdrawn without prior notice.
Upgrades may not apply to suite bookings.
Program is subject to change or cancellation without notice.
Blackout dates and additional terms and conditions may apply.