A method including at least one phase of near-immobilization or immobilization of the line, including the following steps: activation of the motor of at least one upper tensioner and/or of at least one lower tensioner by a control unit, in order to apply to the movement member of the upper tensioner and/or of the lower tensioner, a holding torque to hold the line against its weight; at least occasional control by the control unit of at least one motor of an upper tensioner and/or of a lower tensioner for causing a movement member of the upper tensioner and/or lower tensioner to move or for changing the holding torque applied to the movement member of the upper tensioner and/or of the lower tensioner.